Theorem dnnumch2 42500

Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
dnnumch.f	⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dnnumch.g	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))

Assertion

Ref	Expression
dnnumch2	⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem dnnumch2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.f	. . 3 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2		dnnumch.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		dnnumch.g	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
4	1, 2, 3	dnnumch1 42499	. 2 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
5		f1ofo 6851	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → (𝐹 ↾ 𝑥):𝑥–onto→𝐴)
6		forn 6819	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
8		resss 6011	. . . . . 6 ⊢ (𝐹 ↾ 𝑥) ⊆ 𝐹
9		rnss 5945	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥) ⊆ 𝐹 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
10	8, 9	mp1i 13	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
11	7, 10	eqsstrrd 4021	. . . 4 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
13	12	rexlimdvw 3157	. 2 ⊢ (𝜑 → (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
14	4, 13	mpd 15	1 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3473   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4326  𝒫 cpw 4606   ↦ cmpt 5235  ran crn 5683   ↾ cres 5684  Oncon0 6374  –onto→wfo 6551  –1-1-onto→wf1o 6552  ‘cfv 6553  recscrecs 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398

This theorem is referenced by:  dnnumch3lem  42501  dnnumch3  42502